It's a skipping sort of an affair, rather joyful, and, as its name suggests, it's based on the old-fashioned and not-nearly-so-posh dance called the jig.
But but but...the thing is, although the word gigue sounds quite like th word jig, and the dance looks and sounds like a jig, and in fact is in every way like a jig (except for perhaps tending to be more intellectual and restrained) the similarity of the words seems to be a coincidence.
At least, that's what my Collins dictionary says, and other dictionaries are similarly cautious.
Well, there's etymologists for you.
The word gigue does have some real connections, though - to the gigot, which can either be a leg of lamb or mutton or a leg of mutton sleeve:
drawing by David Ring to a commission from Europeana Fashion
and to the gigolo, who is a man paid to please ladies in various ways, most publicly by dancing with them: though a foxtrot, I should imagine, rather than a jig.
Word To Use Today: gigue. This word comes from French, from the Italian giga, a fiddle. The word jig, surprisingly, no one is quite sure about.